In the development of circuitry and control systems for digital devices, it has long been a problem as to how to sufficiently implement logic circuitry in order to provide control signals to regulate and control associated circuitry and to control a target digital device.
In earlier periods of technology, analog signals and analog circuitry were used to generate analog signals which could be transmitted to analog target devices in order to control their operation. With the advent of digital logic circuitry and with the great flexibility of use of Field Programmable Gate Arrays (FPGAs), and also with the use of Application Specific Integrated Circuits (ASICs), it is now possible that great efficiencies can be provided in using digital busses carrying multiple-bit digital signals which then can be digitally processed in order to provide the desired or required output signals or control signals for target digital devices.
Many of the Input/Output and control signals in digital circuitry will be seen to have mathematical relationships to each other. Thus, with the use of what are called functional relationship generators, which can efficiently be implemented in digital circuitry, it can be found that there is considerably greater efficiency in using various signals and parameters in order to develop a desired output control signal.
The presently described system and method illustrates how mathematical and functional relationships of digital signals in electronic circuitry can be represented by the use of simple, standardized, logic designed elements. These systems and methods can be built into simple, industry-standard programmable logic elements, for example, such as PALs (Programmable Logic Arrays), or Field Programmable Gate Arrays (FPGAs).
Thus, as seen in FIG. 1A, there is indicated a graphical representation of simple linear functional relationship. Here, a value Y (vertical axis) is seen to be a basic function of X (horizontal axis). As X increases or decreases, the value Y increases or decreases in direct proportion.
Now referring to Fig. 1B, there is seen a more generalized linear function. Here, the relationship between Y and X are specified in their most general linear form by a particular equation: EQU Y=a+bX
where a =the intercept value of Y and b =the slope of the line involved.
Thus, it will be seen that when X has a value of 0 (X=0), then Y has a value of "a" (which is the intercept constant).
The constant "b" (slope) controls how quickly Y will change as the value of X changes --that is to say, the slope of the linear line. The greater the "b" value, the steeper the line, and therefore the greater impact each change in X has on the value of Y.
In the simplified case of FIG. 1A, "a" has the value of 0 and "b" has the value of 1. Thus, when X=0, then Y is also 0 (Y=0). Also, in this case, for each unit change in the X, there is a direct unit change in the value of Y. These "linear" relationships and their digital implementation were the subject of the two previously indicated patent applications, which are incorporated by reference.
In the instant case, the focus of the present disclosure will involve the "exponential relationship". FIG. 2 shows the graphical representation of a simple exponential functional relationship. Here, the vertical value, Y, varies "exponentially" with regard to the horizontal variable X. An exponential relationship implies a growing and expanding significance to the increasing values of the independent variable X. That is to say that, each time X increases and grows, its relationship effect on Y increases considerably more. The power of this function is that when X is near to the origin 0, FIG. 2, then changes in X will have a very little effect on the value of Y. However, when X is farther away from the origin point 0 of FIG. 2, (also FIG. 11), then small changes in X will have a very great effect on the value of Y. Such a relationship might be very effective in a device control system to correct for deviations from some standard value at the origin. The further away from the standard (value of X), the greater force (Y) that is needed to be applied to bring it back to a standard control point.
The exponential relationship involved in the present disclosure can be represented as a function of some constant value raised to the power of X. In this particular situation of FIG. 2, the constant "b" is not the slope of the linear functions of FIGS. 1A, 1B, but rather now is used to represent a numerical base which is raised to the X power. In a digital system as seen in FIG. 2, the constant "b" is chosen to be the number "2", since "2" is the base of binary numbers. Thus, Table I below, gives the value of Y for some values of X, showing and illustrating the exponential growth. Thus here, EQU Y=f(b.sup.x)
TABLE 1 ______________________________________ EXPONENTIAL VALUE RELATIONSHIPS change factor X Y = 2.sup.X Y.sub.n to Y.sub.n+1 ______________________________________ 0 1 N/A 1 2 1 2 4 2 3 8 4 4 16 8 5 32 16 6 64 32 7 128 64 8 256 128 ______________________________________